Ridge Length Of Small Stellated Dodecahedron Given Volume Calculator

Formula Used:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \left( \frac{4 \times V}{5 \times (7+3\sqrt{5})} \right)^{\frac{1}{3}} \]

Ridge Length of Small Stellated Dodecahedron:

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1. What is the Ridge Length of Small Stellated Dodecahedron?

The Ridge Length of Small Stellated Dodecahedron is the distance between any inwards directed pyramidal apex and any of its adjacent peak vertex of the Small Stellated Dodecahedron. It is an important geometric measurement in this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Ridge} = \frac{1+\sqrt{5}}{2} \times \left( \frac{4 \times V}{5 \times (7+3\sqrt{5})} \right)^{\frac{1}{3}} \]

Where:

\( l_{Ridge} \) — Ridge Length of Small Stellated Dodecahedron (meters)
\( V \) — Volume of Small Stellated Dodecahedron (cubic meters)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the ridge length based on the volume of the Small Stellated Dodecahedron, incorporating the golden ratio and cubic root relationship.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for understanding the geometric properties and proportions of the Small Stellated Dodecahedron, which has applications in mathematical modeling, architectural design, and crystallography.

4. Using the Calculator

Tips: Enter the volume of the Small Stellated Dodecahedron in cubic meters. The value must be positive and valid. The calculator will compute the corresponding ridge length.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It's a Kepler-Poinsot polyhedron that consists of 12 pentagram faces with the Schläfli symbol {5/2,5}.

Q2: Why does the formula include the golden ratio?
A: The golden ratio (φ = (1+√5)/2) appears naturally in the geometry of pentagonal and dodecahedral shapes.

Q3: What are typical values for ridge length?
A: The ridge length depends on the volume, but for standard-sized polyhedra, it typically ranges from a few centimeters to several meters.

Q4: Can this calculator handle very large volumes?
A: Yes, the calculator can handle a wide range of volume values, though extremely large values may be limited by computational precision.

Q5: Is this formula applicable to other polyhedra?
A: No, this specific formula is derived for the Small Stellated Dodecahedron only. Other polyhedra have different geometric relationships.